Euler's identity

For other meanings, see Euler function (disambiguation)

In complex analysis, Euler's identity is the equation

$e^{i\pi }+1=0\,\!$,

where

$e\,\!$ is the base of the natural logarithm,

$i\,\!$ is the imaginary unit, one of the two complex numbers whose square is negative one (the other is:$-i\,\!$), and

$\pi \,\!$ is Pi, the ratio of the circumference of a circle to its diameter.

Derivation

The equation appears in Leonhard Euler's Introductio, published in Lausanne in 1748. The identity is a special case of Euler's formula from complex analysis, which states that

$e^{ix}=\cos x+i\sin x\,\!$

for any real number $x\,\!$. If $x=\pi \,\!$, then

$e^{i\pi }=\cos \pi +i\sin \pi \,\!$

and since

$\cos \pi =-1\,\!$

and

$\sin \pi =0\,\!$

it follows that

$e^{i\pi }=-1\,\!$

which gives the identity.

Perceptions of the identity

Benjamin Peirce, the noted nineteenth century mathematician and Harvard professor, after proving the identity in a lecture, said, "Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don't know what it means. But we have proved it, and therefore we know it must be the truth."(refactored from Maor)

Many people find this formula remarkable for its mathematical beauty because it links the most fundamental mathematical constants:

• The number 0, the identity element for addition (for all a, a + 0 = 0 + a = a). See Group (mathematics).
• The number 1, the identity element for multiplication (for all a, a × 1 = 1 × a = a).
• The number π is a fundamental constant of trigonometry, Euclidean geometry, and mathematical analysis.
• The number e is a fundamental in connections to the study of logarithms and in calculus (such as in describing growth behaviors, as the solution to the simplest growth equation dy / dx = y with initial condition y(0) = 1 is y = e^x).
• The imaginary unit i (where i² = −1) is a unit in the complex numbers. Introducing this unit yields all non-constant polynomial equations soluble in the field of complex numbers (see fundamental theorem of algebra).

Furthermore, all the most fundamental operators of arithmetic are also present: equality, addition, multiplication and exponentiation.

References

• Maor, Eli, e: The Story of a number, Princeton University Press (May 4 1998), ISBN 0691058547

Notes

Template:Ent Maor, p. 160. Maor cites Edward Kasner and James Newman's, Mathematics and the Imagination, New York: Simon and Schuster (1940), pp. 103–104, as the source for this quote.

See also

• Exponential function

External links

• Proof of Euler's Identity by Julius O. Smith III
• Proof of Euler's Relation by Craig Lewis
• Euler Like equations (Identities which are similar to Euler's Eqn.)